This problem is still open (as well as its analog for \((0)\)  spaces).
The \((n-1)\) -tree comparison provides a necessary condition for \(n\) -point metric spaces, see [1]}.
(The so called matrix inequality turns out to be weaker; see the discussion below.)
This condition is sufficient for the 4-point metric spaces.
It might be still sufficient for 5-point metric spaces,
but not for 6-point metric spaces.
